1 Chapter 02 Continuous Wavelet Transform CWT. 2 Definition of the CWT The continuous-time wavelet...

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Chapter 02Chapter 02Continuous Wavelet Transform CWTContinuous Wavelet Transform CWTChapter 02Chapter 02Continuous Wavelet Transform CWTContinuous Wavelet Transform CWT

22

Definition of the CWTDefinition of the CWTDefinition of the CWTDefinition of the CWT

dtttffsfWsW ss )()(),)((),( ,*

,*

0 , )(, 2 sRsRLf

The continuous-time wavelet transform (CWT)of f(t) with respect to a wavelet (t):

)(2 RL

fW

),)(( sfW

33

Mother WaveletMother WaveletDilation / TranslationDilation / TranslationMother WaveletMother WaveletDilation / TranslationDilation / Translation

s

tsts

2/1

, || )( )( )(0,1 tt Mother Wavelet

s Dilation Scale Translation

0 , )(, 2 sRsRLf

dts

ttfsdtttffsfWsW ss

*2/1

,*

,* )()()(),)((),(

44

}|)(| | :{)( 22

dttfCRfRL

Definition of a mother Wavelet (or Wavelet)Definition of a mother Wavelet (or Wavelet)Definition of a mother Wavelet (or Wavelet)Definition of a mother Wavelet (or Wavelet)

0)(

dtt

A real or complex-value continuous-time function (t)satisfying the following properties, is called a Wavelet:

dtt2

)(

1.

2.

CdC 0 )(

2

3.

Finite energy

Wavelet

Admissibility condition.Sufficient, but not a necessary conditionto obtain the inverse.

55

The Haar Wavelet and The Morlet WaveletThe Haar Wavelet and The Morlet WaveletThe Haar Wavelet and The Morlet WaveletThe Haar Wavelet and The Morlet Wavelet

tet t

2ln

2cos)(

2

otherwise

12

1 1

2

10 1

)( t

t

t

1

-1

1

Haar Morlet

66

Forward / Inverse TransformForward / Inverse TransformForward / Inverse TransformForward / Inverse Transform

dtttffsfWsW ss )()(),)((),( ,*

,*

0 , )(, 2 sRsRLf

Forward

dsdtsWtf s )(),()( ,Inverse

s

tsts

2/1

, || )( Mother Wavelet

s Dilation Scale Translation

77

Admissibility conditionAdmissibility conditionAdmissibility conditionAdmissibility condition

0)(

dtt

It can be shown that square integrable functions (t)satisfying the admissibility condition can be used to first analyze and then reconstruct a signal without loss of information.

CdC 0 )(

2

Admissibility condition.

Sufficient, but not a necessary conditionto obtain the inverse.

The admissibility condition implies that the Fourier transform of (t)vanishes at the zero frequency.

0)(2

A zero at the zero frequency also meansthat the average value of the waveletin the time domain must be zero.

Wavelets must have a band-pass like spectrum.

(t) must be oscillatory,it must be a wave.

88

Regularity conditions - Vanishing momentsRegularity conditions - Vanishing momentsRegularity conditions - Vanishing momentsRegularity conditions - Vanishing moments

The time-bandwidth product of the wavelet transform is the square of the input signal. For most practical applications this is not a desirable property.Therefor one imposes some additional conditions on the wavelet functionsin order to make the wavelet transform decrease quickly with decreasingscale s. These are the regularity conditions and they state that the waveletfunction should have some smoothness and concentration in both time andfrequency domains.Taylor series at t = 0 until order n (let = 0 for simplicity):

)1()0(

)1(!

)0()0,(

0

1)(2/1

0

)(2/1

nOsMfs

nOdts

t

p

tfssW

n

p

pp

p

n

p

pp dtt

p

tM

p

p )(!

pth moment of the wavelet

Moments up to Mn is zero implies that the coefficients of W(s,t) will decay as fast as sn+2 for a smooth signal.

Oscillation + fast decay =Wave + let = Wavelet

99

Dilation / Translation: Haar WaveletDilation / Translation: Haar WaveletDilation / Translation: Haar WaveletDilation / Translation: Haar Wavelet

otherwise

12

1 1

2

10 1

)()(0,1 t

t

tt

Haar

)(0,2 t

1

-1

4

)(0,1 t )(1,2 t

1

-1

4

1

-1

41 2

2

2-1/2 2-1/2

1010

Dilation / Translation: Morlet WaveletDilation / Translation: Morlet WaveletDilation / Translation: Morlet WaveletDilation / Translation: Morlet Wavelet

Morlet

)(0,2 t)(0,1 t )(1,2 t

tet t

2ln

2cos)(

2

1111

CWT - Correlation 1CWT - Correlation 1CWT - Correlation 1CWT - Correlation 1

)()()()()(),(0,,0,, sfss RttfttfsW

)()()()()( *, tytxdttytxR yx

CWT

Cross-correlation

CWT W(s,) is the cross-correlation at lag (shift) between f(t) and the wavelet dilated to scale factor s.

1212


2,

22 ||)(||||)(|| |),(| ttfbaW ba

)()(

||)(||||)(|| |),(|

,

2,

22

tft

ttfbaW

ba

ba

W(a,b) always exists

The global maximum of |W(a,b)| occurs if there is a pair of values (a,b)for which ab(t) = f(t).

Even if this equality does not exists, the global maximum of the real part of W2(a,b) provides a measure of the fit between f(t) and the corresponding ab(t) (se next page).

1313


)],(Re[2||)(|| ||)(|| ||)()(|| 2,

22, baWttfttf baba

The global maximum of the real part of W2(a,b)provides a measure of the fit between f(t) and the corresponding ab(t)

ab(t) closest to f(t) for that value of pair (a,b)for which Re[W(a,b)] is a maximum.

)],(Re[2||)(|| ||)(|| ||)()(|| 2,

22, baWttfttf baba

-ab(t) closest to f(t) for that value of pair (a,b)for which Re[W(a,b)] is a minimum.

1414

CWT - Localization both in time and frequencyCWT - Localization both in time and frequencyCWT - Localization both in time and frequencyCWT - Localization both in time and frequency

The CWT offers time and frequency selectivity;that is, it is able to localize events both in time and in frequency.

Time:The segment of f(t) that influences the value of W(a,b) for any (a,b)is that stretch of f(t) that coinsides with the interval over which ab(t)has the bulk of its energy.This windowing effect results in the time selectivity of the CWT.

Frequency:The frequency selectivity of the CWT is explained using its interpretationas a collection of linear, time-invariant filters with impulse responsesthat are dilations of the mother wavelet reflected about the time axis(se next page).

1515

CWT - Frequency - Filter interpretationCWT - Frequency - Filter interpretationCWT - Frequency - Filter interpretationCWT - Frequency - Filter interpretation

dtxthtxth )()()(*)(Convolution

)(*)(),( *0, bbfbaW a CWT

CWT is the output of a filter with impulse response *ab(-b) and

input f(b).

We have a continuum of filters parameterized by the scale factor a.

1616

CWT - Time and frequency localization 1CWT - Time and frequency localization 1CWT - Time and frequency localization 1CWT - Time and frequency localization 1

dtt

dttt

t2

2

0

)(

)(

dt

dt

2

2

0

)(

)(

dtt

dtttt

t2

220

)(

)()(

dt

dt

2

220

)(

)()(

TimeCenter of mother wavelet

FrequencyCenter of the Fourier transformof mother wavelet

1717


taatata

)()(0,

Time

Frequency

ta

aaa

1

)()(0,

2

1 )()(

productbandwidth -timesmallest thegivesfunctionGaussian

2

1)(

22

2

t

etf

ctaat

Time-bandwidth productis a constant

1818


taatata

)()(0,

Time

Frequency

ta

aaa

1

)()(0,

Small a: CWT resolve events closely spaced in time.Large a: CWT resolve events closely spaced in frequency.

CWT provides better frequency resolution in the lower end of the frequency spectrum.

Wavelet tool a natural tool in the analysis of signals in which rapidlyvarying high-frequency components are superimposed on slowly varyinglow-frequency components (seismic signals, music compositions, …).

1919


t

Time-frequency cells for a,b(t) shown for varied a and fixed b.

a=1/2

a=1

a=2

2020

}|)(| | :{

)2,0(

2

2

dttfCRf

L

2

0

int )(2

1hvor )( dtetfcectf tin

nn

n

nmggdttgtggg

ectg

nmnmnm

tinnn

0 , )()(2

1 ,

)(2

0

itetw )( tinnn etww )(hvor }{

Fourier-serie

Ortonormalebasis-funksjoner

DilationGenerering

Basis-funksjoner Basis-funksjoner for Hilbert rommet Lfor Hilbert rommet L22(0,2(0,2).).Fourier transformasjon.Fourier transformasjon.

Basis-funksjoner Basis-funksjoner for Hilbert rommet Lfor Hilbert rommet L22(0,2(0,2).).Fourier transformasjon.Fourier transformasjon.

2121

Basis-funksjoner Basis-funksjoner for Hilbert rommet Lfor Hilbert rommet L22(R).(R).Wavelet transformasjon.Wavelet transformasjon.

Basis-funksjoner Basis-funksjoner for Hilbert rommet Lfor Hilbert rommet L22(R).(R).Wavelet transformasjon.Wavelet transformasjon.

abab

jjkjkj

kj

fdtttfdta

bttfaabfW

kfWcctf

,,2/1

,,

kj,,

, )()()(||),)((

2

1,

2)(hvor (t))(

Wavelet-serie

Ortonormalebasis-funksjoner

Generering

mlmlkj

jjkj

dxtgtfgf

Zkjktt

,kj,,,

2/,

, )()(,

, )2(2)(

)(t )2(2)(hvor }{ 2/,, ktt jjkjkj

Dilation, translation

a

btatab 2/1

, ||)(

}|)(| | :{

)(

2

2

dttfCRf

RL

2222

Fourier transformertFourier transformertav Wavelet funksjon av Wavelet funksjon Fourier transformertFourier transformertav Wavelet funksjon av Wavelet funksjon

dtetftfFff ti )()}({)(ˆˆ

ibabab eaaa )(ˆ||)(ˆˆ 2/1

;;

Bevis:

ib

uiaib

baui

ti

tiabababab

eaaa

dueeuaa

adueua

dtea

bta

dtettF

)(ˆ||

)(||

)(||

||

)()}({)(ˆˆ

2/1

2/1

)(2/1

2/1

;;;;

2323

CC - Teorem - Teorem CC - Teorem - Teorem

dba

daabgWabfWgfC

2),)((),)((

2424

Teorem Teorem Teorem Teorem

)(ˆ||2

1)(ˆ)(ˆ||

2

1)(ˆ)(ˆ||

2

1

)(ˆ||)(ˆ2

1ˆˆ

2

1),)((

2/12/12/1

2/1;;

bFaafeaaafeaaa

eaaafffabfW

aibib

ibabab

Bevis Bevis Bevis Bevis

)(ˆ)(ˆ)(

)(ˆ)(ˆ)(

axxgxG

axxfxF

a

a

Benytter følgende notasjon:

)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)()(ˆ xfeaxdxxfeaxdxeaxxfdxexFF xixixixiaa

)()(||2

)(ˆ)(ˆ2

1

||2

)(ˆ)(ˆ2

1

||2),)((),)((

22

2

bFbGa

abFbG

a

a

dbbFbGa

adbabgWabfW

aaaa

aa

)(ˆ)(ˆ2

1

||2)(ˆ

2

||)(ˆ

2

||),)((),)((

22/12/1

bFbGa

abG

aabF

aaabgWabfW aaaa

2525

Teorem Teorem Teorem Teorem

dbdya

byygdx

a

bxxf

a

dbgfdbgf

dbdsa

bstgdt

a

bttfadbabgWabfW

abababab

ˆ)(ˆˆ)(ˆ)2(

||

ˆˆˆˆ)2(

1

)()(||),)((),)((

2

1

;;2;;

1

Bevis Bevis Bevis Bevis

)(ˆ)(ˆ)(

)(ˆ)(ˆ)(

axxgxG

axxfxF

a

a

Benytter følgende notasjon:

2626

Binary dilation / Dyadic translationBinary dilation / Dyadic translationBinary dilation / Dyadic translationBinary dilation / Dyadic translation

)(2 RL

)(t

)2()(hvor }{ ,, ktt jkjkj

Binary dilation

Dyadic translationj2

j

k

2

2727

Filtering / CompressionFiltering / CompressionFiltering / CompressionFiltering / Compression

)(2 RL

)(tf ),)(( abfW

Data compression

Remove low W-values

Lowpass-filtering

Replace W-values by 0for low a-values

Highpass-filtering

Replace W-values by 0for high a-values

2828

Inverse Wavelet transformation 1Inverse Wavelet transformation 1Inverse Wavelet transformation 1Inverse Wavelet transformation 1

)(2 RL

)(tf ),)(( abfW

WT

IWT = WT-1

MabfW ),)((

Modifisert

~ Dual

2929

)(2 RL

)(tf ),)(( abfW

WT

a,b a,b R Ra,b a,b R R

0 )(

|)(|

||

|)(ˆ| 2

dtt

dtt

dC

22/1

2

||)],)([(1

)(a

dadb

a

btaabfW

Ctf

R

~

Dual

Basic Wavelet


Condition

Inverse

3030

)(2 RL

)(tf ),)(( abfW

WT

a,b a,b R Ra > 0a > 0a,b a,b R Ra > 0a > 0

0 )(

|)(|

||

|)(ˆ|

||

|)(ˆ|

2

1

0

2

0

2

dtt

dtt

ddC

22/1

2

||)],)([(2

)(a

dadb

a

btaabfW

Ctf

R

~

Dual


Condition

Inverse

3131

)(2 RL

)(tf ),)(( abfW

WT

Condition

Inverse

0 )(

|)(|

||

|)(ˆ| 2

dtt

dtt

dC

22/1

2

||)],)([(1

)(a

dadb

a

btaabfW

Ctf

R

~

Dual

a,b a,b R Ra = 1/2a = 1/2jj

a,b a,b R Ra = 1/2a = 1/2jj

Dyadic Wavelet


3232

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1 Chapter 02 Continuous Wavelet Transform CWT. 2 Definition of the CWT The continuous-time wavelet...

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Transcript of 1 Chapter 02 Continuous Wavelet Transform CWT. 2 Definition of the CWT The continuous-time wavelet...